This paper examines multiplayer symmetric constant-sum games with more than two players in a competitive setting, including examples like Mahjong, Poker, and various board and video games. In contrast to two-player zero-sum games, equilibria in multiplayer games are neither unique nor non-exploitable, failing to provide meaningful guarantees when competing against opponents who play different equilibria or non-equilibrium strategies. This gives rise to a series of long-lasting fundamental questions in multiplayer games regarding suitable objectives, solution concepts, and principled algorithms. This paper takes an initial step towards addressing these challenges by focusing on the natural objective of equal share -- securing an expected payoff of C/n in an n-player symmetric game with a total payoff of C. We rigorously identify the theoretical conditions under which achieving an equal share is tractable and design a series of efficient algorithms, inspired by no-regret learning, that provably attain approximate equal share across various settings. Furthermore, we provide complementary lower bounds that justify the sharpness of our theoretical results. Our experimental results highlight worst-case scenarios where meta-algorithms from prior state-of-the-art systems for multiplayer games fail to secure an equal share, while our algorithm succeeds, demonstrating the effectiveness of our approach.

翻译：本文研究竞争环境下超过两名玩家的多人对称常和博弈，包括麻将、扑克以及各类棋盘与电子游戏等实例。与双人零和博弈不同，多人博弈中的均衡既非唯一也非不可利用，当对手采用不同均衡或非均衡策略时，这些均衡无法提供有意义的性能保证。这引发了关于多人博弈中合适目标、解概念与原理性算法的一系列长期存在的根本性问题。本文通过聚焦于"公平份额"这一自然目标——在总收益为C的n人对称博弈中确保获得C/n的期望收益——迈出了应对这些挑战的第一步。我们严格界定了实现公平份额在理论上的可行条件，并受无悔学习思想启发，设计了一系列高效算法，可证明在多种设定下获得近似公平份额。此外，我们给出了互补的下界结果，证明了理论结论的紧致性。实验结果表明，在现有最先进多人博弈系统元算法无法保障公平份额的最坏情况下，我们的算法仍能成功实现该目标，验证了所提方法的有效性。